WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.260 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.260 --> 00:00:21.730 at ocw.mit.edu 00:00:21.730 --> 00:00:25.160 MEHRAN KARDAR: OK, let's start. 00:00:25.160 --> 00:00:31.730 So in this class, we focused mostly 00:00:31.730 --> 00:00:41.150 on having some slab of material and having some configuration 00:00:41.150 --> 00:00:45.310 of some kind of a field inside. 00:00:45.310 --> 00:00:49.720 And we said that, basically, we are 00:00:49.720 --> 00:00:52.810 going to be interested close to, let's say, 00:00:52.810 --> 00:00:56.950 phase transition and some quantity that changes 00:00:56.950 --> 00:00:57.950 at the phase transition. 00:00:57.950 --> 00:01:01.330 We are interested in figuring out the singularities 00:01:01.330 --> 00:01:03.350 associated with that. 00:01:03.350 --> 00:01:06.710 And we can coarse grain. 00:01:06.710 --> 00:01:12.470 Once we have coarse grained, we have the field m, potentially 00:01:12.470 --> 00:01:16.950 a vector, that is characterized throughout this material. 00:01:16.950 --> 00:01:20.960 So it's a field that's function of x. 00:01:20.960 --> 00:01:28.780 And by integrating out over a lot of degrees of freedom, 00:01:28.780 --> 00:01:32.170 we can focus on the probability of finding 00:01:32.170 --> 00:01:37.080 different configurations of this field. 00:01:37.080 --> 00:01:40.150 And this probability we constructed 00:01:40.150 --> 00:01:46.010 on the basis of a number of simple assumptions 00:01:46.010 --> 00:01:50.780 such as locality, which implied that we would write 00:01:50.780 --> 00:01:56.360 this probability as a product of contributions 00:01:56.360 --> 00:02:01.590 of different parts, which in the exponent becomes an integral. 00:02:01.590 --> 00:02:05.520 And then we would put within this all kinds 00:02:05.520 --> 00:02:09.560 of things that are consistent with the symmetries 00:02:09.560 --> 00:02:10.599 of the problem. 00:02:10.599 --> 00:02:13.370 So if, for example, this is a field that 00:02:13.370 --> 00:02:16.090 is invariant on the rotations, we 00:02:16.090 --> 00:02:22.735 would be having terms such as m squared, m to the fourth, 00:02:22.735 --> 00:02:23.580 and so forth. 00:02:26.630 --> 00:02:28.620 But the interesting thing was that, of course, 00:02:28.620 --> 00:02:32.380 there is some interaction with the neighborhoods. 00:02:32.380 --> 00:02:36.520 Those neighborhood interactions we can, in the continuum limit, 00:02:36.520 --> 00:02:40.670 inclement by putting terms that are proportional to radiant 00:02:40.670 --> 00:02:42.540 of m and so forth. 00:02:50.240 --> 00:02:52.280 So there is a lot of things that you 00:02:52.280 --> 00:02:55.620 could put consistent with symmetry 00:02:55.620 --> 00:02:58.550 and presumably be as general as possible. 00:02:58.550 --> 00:03:00.880 You could have this and the terms 00:03:00.880 --> 00:03:04.620 in this coefficients of this expansion would 00:03:04.620 --> 00:03:08.260 be these phenomenological parameters characterizing 00:03:08.260 --> 00:03:11.230 this probability, function of all kinds 00:03:11.230 --> 00:03:13.090 of microscopic degrees of freedom, 00:03:13.090 --> 00:03:15.310 as well as microscopic constraints 00:03:15.310 --> 00:03:19.140 such as temperature, pressures, et cetera. 00:03:19.140 --> 00:03:27.390 Now the question that we have is if I start with a system, 00:03:27.390 --> 00:03:30.190 let's say a configuration where everybody's pointing up 00:03:30.190 --> 00:03:33.990 or some other configuration that is not the equilibrium 00:03:33.990 --> 00:03:36.930 configuration, how does the probability 00:03:36.930 --> 00:03:41.290 evolve to become something like this? 00:03:41.290 --> 00:03:44.970 Now we are interested therefore in m 00:03:44.970 --> 00:03:49.030 that is a function of position and time. 00:03:49.030 --> 00:03:54.180 And since I want to use t for time, this coefficient of m 00:03:54.180 --> 00:03:57.366 squared that we were previously calling t, 00:03:57.366 --> 00:04:02.250 I will indicate by r, OK? 00:04:02.250 --> 00:04:04.470 There are various types of dynamics that you 00:04:04.470 --> 00:04:07.020 can look at for this problem. 00:04:07.020 --> 00:04:09.562 I will look at the class that is dissipative. 00:04:15.880 --> 00:04:20.100 And its inspiration is the Brownian motion 00:04:20.100 --> 00:04:22.980 that we discussed last time where 00:04:22.980 --> 00:04:27.850 we saw that when you put a particle in a fluid 00:04:27.850 --> 00:04:31.910 to a very good approximation when the fluid is viscous, 00:04:31.910 --> 00:04:34.880 it is the velocity that is proportional to the force 00:04:34.880 --> 00:04:38.600 and you can't ignore inertial effects such as mass times 00:04:38.600 --> 00:04:41.330 acceleration and you write an equation 00:04:41.330 --> 00:04:45.930 that is linear in position. 00:04:45.930 --> 00:04:48.210 Velocity is the linear derivative 00:04:48.210 --> 00:04:50.180 of position, which is the variable that 00:04:50.180 --> 00:04:52.130 is of interest to you. 00:04:52.130 --> 00:04:55.000 So here the variable that is of interest to us, 00:04:55.000 --> 00:04:59.010 is this magnetization that is changing as a function of time. 00:04:59.010 --> 00:05:02.100 And the equation that we write down 00:05:02.100 --> 00:05:09.400 is the derivative of this field with respect to time. 00:05:09.400 --> 00:05:14.650 And again, for the Brownian motion, 00:05:14.650 --> 00:05:18.400 the velocity was proportional to the force. 00:05:18.400 --> 00:05:22.380 The constant of proportionality was some kind of a mobility 00:05:22.380 --> 00:05:24.340 that you would have in the fluid. 00:05:24.340 --> 00:05:27.530 So continuing with that inspiration, 00:05:27.530 --> 00:05:29.740 let's put some kind of a mobility here. 00:05:29.740 --> 00:05:32.830 I should really put a vector field here, but just 00:05:32.830 --> 00:05:38.420 for convenience, let's just focus on one component 00:05:38.420 --> 00:05:40.830 and see what happens. 00:05:40.830 --> 00:05:44.730 Now what is the force? 00:05:44.730 --> 00:05:49.670 Presumably, each location individually 00:05:49.670 --> 00:05:52.570 feels some kind of a force. 00:05:52.570 --> 00:05:56.660 And typically when we had the Brownian particle, the force 00:05:56.660 --> 00:05:58.420 we were getting from the derivative 00:05:58.420 --> 00:06:02.125 of the potential with respect to the [? aviation ?] 00:06:02.125 --> 00:06:06.390 of the position, the field that we are interested. 00:06:06.390 --> 00:06:13.030 So the analog of our potential is the energy 00:06:13.030 --> 00:06:17.280 that we have over here, and in the same sense 00:06:17.280 --> 00:06:21.430 that we want this effective potential, if you like, 00:06:21.430 --> 00:06:24.760 to govern the equilibrium behavior-- 00:06:24.760 --> 00:06:28.050 and again, recall that for the case of the Brownian particle, 00:06:28.050 --> 00:06:31.670 eventually the probability was related to the potential by e 00:06:31.670 --> 00:06:36.450 to the minus beta v. This is the analog of beta v, now 00:06:36.450 --> 00:06:38.770 considered in the entire field. 00:06:38.770 --> 00:06:45.740 So the analog of the force is a derivative of this beta H, 00:06:45.740 --> 00:06:53.600 so this would be our beta H with respect to the variable 00:06:53.600 --> 00:06:55.590 that I'm trying to change. 00:06:58.930 --> 00:07:05.325 And since I don't have just one variable, but a field, 00:07:05.325 --> 00:07:08.350 the analog of the derivative becomes this functional 00:07:08.350 --> 00:07:09.980 derivative. 00:07:09.980 --> 00:07:14.160 And in the same sense that the Brownian particle, 00:07:14.160 --> 00:07:17.540 the Brownian motion, is trying to pull you 00:07:17.540 --> 00:07:20.470 towards the minimum of the potential, 00:07:20.470 --> 00:07:24.450 this is an equation that, if I kind of stop over here, 00:07:24.450 --> 00:07:28.090 tries to put the particle towards the minimum 00:07:28.090 --> 00:07:33.000 of this beta H. 00:07:33.000 --> 00:07:36.530 Now the reason that the Brownian particle didn't go and stick 00:07:36.530 --> 00:07:40.220 at one position, which was the minimum, but fluctuated, 00:07:40.220 --> 00:07:43.310 was of course, that we added the random force. 00:07:43.310 --> 00:07:46.970 So there is some kind of an analog of a random force that 00:07:46.970 --> 00:07:51.157 we can put either in front of [? mu ?] or imagine that if we 00:07:51.157 --> 00:07:56.100 added it to [? mu, ?] and we put it over here. 00:07:56.100 --> 00:07:59.740 Now for the case of the Brownian particle, 00:07:59.740 --> 00:08:04.930 the assumption was that if I evaluate eta at some time, 00:08:04.930 --> 00:08:08.860 and eta at another time t2 and t1, 00:08:08.860 --> 00:08:17.520 that this was related to 2D delta function t1 minus t2. 00:08:17.520 --> 00:08:23.840 Now of course here, at each location, I have a noise term. 00:08:23.840 --> 00:08:28.400 So this noise carries an index, which 00:08:28.400 --> 00:08:31.980 indicates the position, which is, of course, a vector in D 00:08:31.980 --> 00:08:34.309 dimension in principal. 00:08:34.309 --> 00:08:38.610 And there is no reason to imagine that the noise here 00:08:38.610 --> 00:08:42.210 that comes from all kinds of microscopic degrees of freedom 00:08:42.210 --> 00:08:45.310 that we have integrated out should 00:08:45.310 --> 00:08:48.650 have correlation with the noise at some other point. 00:08:48.650 --> 00:08:54.520 So the simplest assumption is to also put the delta function 00:08:54.520 --> 00:08:58.710 in the positions, OK? 00:08:58.710 --> 00:09:02.340 So if I take this beta H that I have over there 00:09:02.340 --> 00:09:05.530 and take the functional derivative, what do I get? 00:09:05.530 --> 00:09:17.110 I will get that the m of xt by dt is the function of-- oops, 00:09:17.110 --> 00:09:20.650 and I forgot to put a the minus sign here. 00:09:20.650 --> 00:09:23.290 The force is minus the potential, 00:09:23.290 --> 00:09:26.100 and this is basically going down the gradient, 00:09:26.100 --> 00:09:28.060 so I need to put that. 00:09:28.060 --> 00:09:31.530 So I have to take a derivative of this. 00:09:31.530 --> 00:09:33.765 First of all I can take a derivative with respect 00:09:33.765 --> 00:09:35.780 to m itself. 00:09:35.780 --> 00:09:40.055 I will get minus rm. 00:09:44.640 --> 00:09:48.840 Well, actually, let's put the minus out front. 00:09:48.840 --> 00:09:52.580 And then I will get the derivative of m 00:09:52.580 --> 00:09:56.280 to the fourth, which is [? 4umq, ?] all kinds of terms 00:09:56.280 --> 00:09:57.790 like this. 00:09:57.790 --> 00:10:01.110 Then the terms that come from the derivatives 00:10:01.110 --> 00:10:05.650 and taking the derivative with respect to gradient 00:10:05.650 --> 00:10:08.400 will give me K times the gradient. 00:10:08.400 --> 00:10:10.780 But then in the functional derivative, 00:10:10.780 --> 00:10:13.370 I have to take another derivative, 00:10:13.370 --> 00:10:17.320 converting this to minus K log Gaussian of m. 00:10:17.320 --> 00:10:20.050 And then I would have L the fourth derivative 00:10:20.050 --> 00:10:24.310 from the next term and so forth. 00:10:24.310 --> 00:10:26.330 And on top of everything else, I will 00:10:26.330 --> 00:10:31.080 have the noise who's statistics I have indicated above. 00:10:34.650 --> 00:10:37.880 So this entity, this equation, came 00:10:37.880 --> 00:10:42.820 from the Landau-Ginzburg model, and it 00:10:42.820 --> 00:10:51.325 is called a time dependent Landau-Ginzburg equation. 00:10:54.540 --> 00:11:02.120 And so would be the analog of the Brownian type of equation, 00:11:02.120 --> 00:11:05.810 now for an entire field. 00:11:05.810 --> 00:11:08.820 Now we're going to have a lot of difficulty 00:11:08.820 --> 00:11:12.750 in the short amount of time that is left to us to deal 00:11:12.750 --> 00:11:15.320 with this nonlinear equation, so we 00:11:15.320 --> 00:11:18.160 are going to do the same thing that we did in order 00:11:18.160 --> 00:11:22.300 to capture the properties of the Landau-Ginzburg model 00:11:22.300 --> 00:11:23.720 qualitatively. 00:11:23.720 --> 00:11:26.410 Which is to ignore nonlinearities, 00:11:26.410 --> 00:11:30.000 so it's a kind of Gaussian version of the model. 00:11:30.000 --> 00:11:33.810 So what we did was we linearized. 00:11:33.810 --> 00:11:34.720 AUDIENCE: Question. 00:11:34.720 --> 00:11:35.960 MEHRAN KARDAR: Yes? 00:11:35.960 --> 00:11:38.690 AUDIENCE: Why does the sign in front of the K term 00:11:38.690 --> 00:11:40.830 change relative to the others? 00:11:40.830 --> 00:11:48.140 MEHRAN KARDAR: OK, so when you have the function 0 of a field 00:11:48.140 --> 00:11:56.010 gradient, et cetera, you can show 00:11:56.010 --> 00:12:01.510 that the functional derivative, the first term 00:12:01.510 --> 00:12:06.940 is the ordinary type of derivative. 00:12:06.940 --> 00:12:12.152 And then if you think about the variations that are carrying m 00:12:12.152 --> 00:12:17.870 and with the gradient and you write it as m plus delta m 00:12:17.870 --> 00:12:21.660 and then make sure that you take the delta m outside, 00:12:21.660 --> 00:12:26.280 you need to do an integration by part that changes the sign. 00:12:26.280 --> 00:12:28.566 And the next term would be the gradient 00:12:28.566 --> 00:12:32.760 of the phi by the [? grad ?] m, and the next term 00:12:32.760 --> 00:12:35.550 would be log [INAUDIBLE] of [INAUDIBLE] gradient 00:12:35.550 --> 00:12:38.176 of m squared and so forth. 00:12:38.176 --> 00:12:43.436 It alternates and so by explicitly calculating 00:12:43.436 --> 00:12:45.580 the difference of two functionals 00:12:45.580 --> 00:12:51.320 with this integration evaluated at m and m plus delta m 00:12:51.320 --> 00:12:55.102 and pulling out everything that is proportional to delta m 00:12:55.102 --> 00:12:56.910 you can prove these expressions. 00:13:00.920 --> 00:13:05.880 So we linearize this equation, which I've already done that. 00:13:05.880 --> 00:13:09.980 I cross out the mq term and any other nonlinear terms, 00:13:09.980 --> 00:13:12.630 so I only keep the linear term. 00:13:12.630 --> 00:13:16.065 And then I do a Fourier transform. 00:13:21.330 --> 00:13:27.660 So basically I switch from the position representation 00:13:27.660 --> 00:13:32.620 to the Fourier transform that's called m tilde of q, I think. 00:13:32.620 --> 00:13:35.770 x is replaced with q. 00:13:35.770 --> 00:13:42.220 And then the equation for the field m linears. 00:13:42.220 --> 00:13:46.900 The form separates out into independent equations 00:13:46.900 --> 00:13:51.660 for the components that are characterized by q. 00:13:51.660 --> 00:13:55.200 So the Fourier transform of the left hand side is just this. 00:13:55.200 --> 00:13:58.460 Fourier transform of the right hand side 00:13:58.460 --> 00:14:05.660 will give me minus mu r plus K q squared. 00:14:05.660 --> 00:14:10.140 The derivative of [INAUDIBLE] will give me a minus q squared. 00:14:10.140 --> 00:14:15.290 The derivative of the next term L q to the forth, et cetera. 00:14:15.290 --> 00:14:17.776 And I've only kept the linear term, 00:14:17.776 --> 00:14:21.510 so I have m tilde of q and t. 00:14:21.510 --> 00:14:23.580 And then I have the Fourier transforms 00:14:23.580 --> 00:14:27.000 of the noise eta of x and t. 00:14:27.000 --> 00:14:29.820 They [? call it ?] eta tilde of q [? m. ?] 00:14:33.590 --> 00:14:39.870 So you can see that each mode satisfy 00:14:39.870 --> 00:14:42.940 a separate linear equation. 00:14:45.570 --> 00:14:49.290 So this equation is actually very easy 00:14:49.290 --> 00:14:56.960 to solve for any linear equation m tilde of q and t. 00:14:56.960 --> 00:15:00.230 If I didn't have the noise, so I would 00:15:00.230 --> 00:15:05.360 start with some value at t [? close ?] to 0, 00:15:05.360 --> 00:15:09.940 and that value would decay exponentially 00:15:09.940 --> 00:15:14.660 in the characteristic time that I will call tao of q. 00:15:14.660 --> 00:15:26.980 And 1 over tao of q is simply this mu r plus K q2 00:15:26.980 --> 00:15:30.860 and so forth. 00:15:30.860 --> 00:15:34.920 Now once you have noise, essentially each one 00:15:34.920 --> 00:15:39.180 of these noises acts like an initial condition. 00:15:39.180 --> 00:15:43.300 And so the full answer is an integral 00:15:43.300 --> 00:15:48.470 over all of these noises from 0 tilde time t of interest. 00:15:48.470 --> 00:15:54.260 Dt prime the noise that occurs at a time t prime, 00:15:54.260 --> 00:15:57.515 and the noise that occurs at time t [? trime ?] relaxes 00:15:57.515 --> 00:16:06.350 with this into the minus t minus t prime tao of q. 00:16:06.350 --> 00:16:12.420 So that's the solution to that linear noisy equations, 00:16:12.420 --> 00:16:14.320 basically [? sequencing ?] [INAUDIBLE]. 00:16:17.970 --> 00:16:24.330 So one of the things that we now see 00:16:24.330 --> 00:16:33.630 is that essentially the different Fourier components 00:16:33.630 --> 00:16:35.080 of the field. 00:16:35.080 --> 00:16:38.150 Each one of them is independently 00:16:38.150 --> 00:16:41.710 relaxing to something, and each one of them 00:16:41.710 --> 00:16:46.690 has a characteristic relaxation time. 00:16:46.690 --> 00:16:54.360 As I go towards smaller and smaller values of q, 00:16:54.360 --> 00:16:59.670 this rate becomes smaller, and the relaxation time 00:16:59.670 --> 00:17:01.580 becomes larger. 00:17:01.580 --> 00:17:03.880 So essentially shortwave wavelength modes 00:17:03.880 --> 00:17:07.630 that correspond to large q, they relax first. 00:17:07.630 --> 00:17:11.280 Longer wavelength modes will relax later on. 00:17:11.280 --> 00:17:19.480 And you can see that the largest relaxation time, tao max, 00:17:19.480 --> 00:17:25.243 corresponds to q equals to 0 is simply 1 over [? nu ?] r. 00:17:28.349 --> 00:17:37.270 So I can pluck this to a max as a function of r, 00:17:37.270 --> 00:17:41.200 and again in this theory, the Gaussian theory, 00:17:41.200 --> 00:17:45.410 we saw only makes sense as long as all is positive. 00:17:45.410 --> 00:17:49.410 So I have to look only on the positive axis. 00:17:49.410 --> 00:17:53.060 And I find that the relaxation time 00:17:53.060 --> 00:17:57.060 for the entire system for the longest wavelength 00:17:57.060 --> 00:18:02.310 actually diverges as r goes to 0. 00:18:02.310 --> 00:18:05.560 And recall that r, in our perspective, 00:18:05.560 --> 00:18:09.320 is really something that is proportional to T minus Tc. 00:18:12.260 --> 00:18:16.800 So basically we find that as we are approaching 00:18:16.800 --> 00:18:20.810 the critical point, the time it takes 00:18:20.810 --> 00:18:23.850 for the entirety of the system or the longest 00:18:23.850 --> 00:18:30.820 wavelength [? modes ?] to relax diverges as 1 over T minus Tc. 00:18:30.820 --> 00:18:35.010 There's an exponent that shows this, so called, 00:18:35.010 --> 00:18:36.660 critical slowing down. 00:18:43.950 --> 00:18:46.400 Yes? 00:18:46.400 --> 00:18:48.200 AUDIENCE: In principal, why couldn't you 00:18:48.200 --> 00:18:53.930 have that r becomes negative if you restrict your range of q 00:18:53.930 --> 00:18:58.930 to be outside of some value and not go arbitrarily 00:18:58.930 --> 00:19:01.380 close to the origin. 00:19:01.380 --> 00:19:05.230 MEHRAN KARDAR: You could, but what's the physics of that? 00:19:05.230 --> 00:19:06.056 See. 00:19:06.056 --> 00:19:08.550 AUDIENCE: I'm wondering if there is or is that not needed. 00:19:08.550 --> 00:19:09.300 MEHRAN KARDAR: No. 00:19:09.300 --> 00:19:14.360 OK, so the physics of that could be that you have a system that 00:19:14.360 --> 00:19:21.950 has some finite size, [? l. ?] Then the largest q that you 00:19:21.950 --> 00:19:27.030 could have [? or ?] the smallest q that you could have would be 00:19:27.030 --> 00:19:31.680 the [? 1/l. ?] So in principal, for that you can go slightly 00:19:31.680 --> 00:19:32.610 negative. 00:19:32.610 --> 00:19:37.180 You still cannot go too negative because ultimately this will 00:19:37.180 --> 00:19:39.490 overcome that. 00:19:39.490 --> 00:19:42.020 But again, we are interested in singularities 00:19:42.020 --> 00:19:46.884 that we kind of know arise in the limit of [INAUDIBLE]. 00:19:52.850 --> 00:19:56.150 I also recall, there is a time that we see as diverging 00:19:56.150 --> 00:19:57.810 as r goes to 0. 00:19:57.810 --> 00:20:02.060 Of course, we identified before a correlation length 00:20:02.060 --> 00:20:06.380 from balancing these two terms, and the correlation length 00:20:06.380 --> 00:20:10.540 is square root of K over r, which is again proportional 00:20:10.540 --> 00:20:15.500 to T minus Tc and diverges with a square root singularity. 00:20:15.500 --> 00:20:23.640 So we can see that this tao max is actually related 00:20:23.640 --> 00:20:30.622 to the psi squared over Nu K. 00:20:30.622 --> 00:20:34.150 And it is also related towards this. 00:20:34.150 --> 00:20:47.610 We can see that our tao of q, basically, if q is large 00:20:47.610 --> 00:20:54.980 such that qc is larger than 1 and the characteristic time 00:20:54.980 --> 00:21:03.340 is going to be 1 over [? nu ?] K times the inverse of q squared. 00:21:03.340 --> 00:21:07.390 And the inverse of q is something like a wavelength. 00:21:07.390 --> 00:21:12.290 Whereas, ultimately, this saturates for qc what is less 00:21:12.290 --> 00:21:18.440 than 1 [? long ?] wavelengths to c squared over [? nu ?] K. 00:21:18.440 --> 00:21:26.230 So basically you see things at very short range, 00:21:26.230 --> 00:21:29.040 at length scales that are much less than the correlation 00:21:29.040 --> 00:21:33.760 length of the system, that the characteristic time will 00:21:33.760 --> 00:21:37.475 depend on the length scale that you are looking at squared. 00:21:41.130 --> 00:21:48.070 Now you have seen times scaling as length squared 00:21:48.070 --> 00:21:51.870 from diffusion, so essentially this 00:21:51.870 --> 00:21:56.256 is some kind of a manifestation of diffusion, 00:21:56.256 --> 00:22:01.620 but as you perturb the system, let's say at short distances, 00:22:01.620 --> 00:22:03.930 there's some equilibrium system. 00:22:03.930 --> 00:22:07.120 Let's say we do some perturbation to it 00:22:07.120 --> 00:22:09.790 at some point, and that perturbation 00:22:09.790 --> 00:22:14.940 will start to expand diffusively until it reaches 00:22:14.940 --> 00:22:18.050 the size of the correlation length, at which point 00:22:18.050 --> 00:22:21.050 it stops because essentially correlation length is 00:22:21.050 --> 00:22:25.390 an individual block that doesn't know about individual blocks, 00:22:25.390 --> 00:22:26.900 so the influence does not last. 00:22:30.480 --> 00:22:33.380 So quite generally, what you find-- 00:22:33.380 --> 00:22:36.480 so we solved the linearized version 00:22:36.480 --> 00:22:38.450 of the Landau-Ginzburg model, but we 00:22:38.450 --> 00:22:43.960 know that, say, the critical behaviors for the divergence 00:22:43.960 --> 00:22:47.040 of the correlation length that is predicted here 00:22:47.040 --> 00:22:49.030 is not correct in three dimensions, 00:22:49.030 --> 00:22:50.830 things get modified. 00:22:50.830 --> 00:22:54.580 So these kind of exponents that come from diffusion also 00:22:54.580 --> 00:22:56.690 gets modified. 00:22:56.690 --> 00:23:01.640 And quite generally, you find that the relaxation time 00:23:01.640 --> 00:23:09.610 of a mode of wavelength q is going to behave something 00:23:09.610 --> 00:23:21.140 like wavelength, which is 1 over q, 00:23:21.140 --> 00:23:27.710 rather than squared as some exponent z. 00:23:27.710 --> 00:23:33.190 And then there is some function of the product 00:23:33.190 --> 00:23:34.990 of the wavelength you are looking at 00:23:34.990 --> 00:23:39.020 and the correlation length so that you will cross over 00:23:39.020 --> 00:23:42.260 from one behavior to another behavior 00:23:42.260 --> 00:23:44.280 as you are looking at length scales that 00:23:44.280 --> 00:23:46.050 are smaller than the correlation length 00:23:46.050 --> 00:23:49.090 or larger than the correlation length. 00:23:49.090 --> 00:23:52.600 And to get what this exponent z is, 00:23:52.600 --> 00:23:56.850 you have to do study of the nonlinear model 00:23:56.850 --> 00:23:58.950 in the same sense that, in order to get 00:23:58.950 --> 00:24:01.350 the correction to the exponent [? nu ?], 00:24:01.350 --> 00:24:03.500 we had to do epsilon expansion. 00:24:03.500 --> 00:24:05.115 You have to do something similar, 00:24:05.115 --> 00:24:10.970 and you'll fine that at higher changes, 00:24:10.970 --> 00:24:15.240 it goes like some correction that does not actually 00:24:15.240 --> 00:24:18.580 start at order of epsilon but at order of epsilon squared. 00:24:18.580 --> 00:24:24.520 But there's essentially some modification 00:24:24.520 --> 00:24:27.920 of the qualitative behavior that we 00:24:27.920 --> 00:24:33.180 can ascribe to the fusion of independent modes 00:24:33.180 --> 00:24:37.190 exists quite generally and universal exponents 00:24:37.190 --> 00:24:40.700 different from [? to ?] will emerge from that. 00:24:45.600 --> 00:24:51.900 Now it turns out that this is not the end of the story 00:24:51.900 --> 00:24:54.820 because we have seen that the same probability 00:24:54.820 --> 00:25:00.650 distribution can describe a lot of different systems. 00:25:00.650 --> 00:25:04.910 Let's say the focus on the case of n equals to 1. 00:25:08.680 --> 00:25:11.760 So then this Landau-Ginzburg that I described 00:25:11.760 --> 00:25:15.680 for you can describe, let's say, the Ising model, 00:25:15.680 --> 00:25:24.040 which describes magnetizations that lie 00:25:24.040 --> 00:25:26.700 along the particular direction. 00:25:26.700 --> 00:25:31.040 So that it can also describe liquid gas phenomena 00:25:31.040 --> 00:25:34.480 where the order parameter is the difference in density, 00:25:34.480 --> 00:25:37.760 if you like, between the liquid and the gas. 00:25:37.760 --> 00:25:40.970 Yet another example that it describes 00:25:40.970 --> 00:25:43.650 is the mixing of an alloy. 00:25:50.110 --> 00:25:58.380 So let's, for example, imagine brass that has a composition 00:25:58.380 --> 00:26:01.550 x that goes between 0 and 1. 00:26:01.550 --> 00:26:04.345 On one end, let's say you have entirely copper 00:26:04.345 --> 00:26:08.250 and on the other and you have entirely zinc. 00:26:08.250 --> 00:26:11.890 And so this is how you make brass as an alloy. 00:26:11.890 --> 00:26:16.580 And what my other axis is is the temperature. 00:26:16.580 --> 00:26:21.690 What you find is that there is some kind of phase diagram 00:26:21.690 --> 00:26:29.920 such that you get a nice mixture of copper and zinc 00:26:29.920 --> 00:26:32.180 only if you are at high temperatures, 00:26:32.180 --> 00:26:34.070 whereas if you are at low temperature, 00:26:34.070 --> 00:26:37.660 you basically will separate into chunks 00:26:37.660 --> 00:26:40.950 that are rich copper and chunks that are rich in zinc. 00:26:40.950 --> 00:26:44.710 And you'll have a critical demixing 00:26:44.710 --> 00:26:50.560 point, which has exactly the same properties as the Ising 00:26:50.560 --> 00:26:51.500 model. 00:26:51.500 --> 00:26:54.760 For example, this curve will be characterized 00:26:54.760 --> 00:26:56.610 with an exponent beta, which would 00:26:56.610 --> 00:26:59.811 be the beta of the Ising mode. 00:26:59.811 --> 00:27:05.410 And in particular, if I were to take someplace 00:27:05.410 --> 00:27:10.080 in the vicinity of this and try to write down a probability 00:27:10.080 --> 00:27:13.200 distribution, that probability distribution 00:27:13.200 --> 00:27:15.960 would be exactly what I have over there where 00:27:15.960 --> 00:27:22.470 m is, let's say, the difference between the two types of alloys 00:27:22.470 --> 00:27:28.420 that I have compared to each other over here. 00:27:28.420 --> 00:27:32.060 So this is related to 2x minus 1 or something like that. 00:27:34.880 --> 00:27:40.280 So as you go across your piece of material close to here, 00:27:40.280 --> 00:27:42.190 there will be compositional variations 00:27:42.190 --> 00:27:45.540 that are described by that. 00:27:45.540 --> 00:27:51.620 So the question is, I know exactly what the probability 00:27:51.620 --> 00:27:56.120 distribution is for this system to be an equilibrium given 00:27:56.120 --> 00:27:58.910 this choice of m. 00:27:58.910 --> 00:28:03.020 Again, with some set of parameters, R, U, et cetera. 00:28:03.020 --> 00:28:07.730 The question is-- is the dynamics again described 00:28:07.730 --> 00:28:10.350 by the same equation? 00:28:10.350 --> 00:28:12.810 And the answer is no. 00:28:12.810 --> 00:28:15.640 The same probability of distribution 00:28:15.640 --> 00:28:19.430 can describe-- or can be obtained 00:28:19.430 --> 00:28:21.250 with very different dynamics. 00:28:21.250 --> 00:28:23.640 And in particular, what is happening 00:28:23.640 --> 00:28:28.970 in the system is that, if I integrate this quantity m 00:28:28.970 --> 00:28:34.920 across the system, I will get the total number of 1 00:28:34.920 --> 00:28:38.750 minus the other, which is what is given to you, 00:28:38.750 --> 00:28:41.530 and it does not change as a function of time. 00:28:41.530 --> 00:28:50.585 d by dt of this quantity is 0. 00:28:50.585 --> 00:28:53.058 It cannot change. 00:28:53.058 --> 00:28:53.970 OK. 00:28:53.970 --> 00:28:59.300 Whereas the equation that I have written over here, 00:28:59.300 --> 00:29:03.630 in principle, locally, I can, by adding the noise 00:29:03.630 --> 00:29:05.740 or by bringing things from the neighborhood, 00:29:05.740 --> 00:29:08.135 I can change the value of m. 00:29:08.135 --> 00:29:09.320 I cannot do that. 00:29:09.320 --> 00:29:15.000 So this process of the relaxation that would go 00:29:15.000 --> 00:29:19.100 on in data graphs cannot be described by the time dependent 00:29:19.100 --> 00:29:22.270 on the Landau-Ginzburg equation because you have this 00:29:22.270 --> 00:29:23.652 conservation here. 00:29:28.080 --> 00:29:30.260 OK. 00:29:30.260 --> 00:29:34.050 So what should we do? 00:29:34.050 --> 00:29:39.270 Well, when things are conserved, like, say, 00:29:39.270 --> 00:29:42.580 as the gas particles move in this fluid, 00:29:42.580 --> 00:29:45.460 and if I'm interested in the number of particles 00:29:45.460 --> 00:29:48.390 in some cube, then the change in the number 00:29:48.390 --> 00:29:51.050 of particles in some cube in this room 00:29:51.050 --> 00:29:54.310 is related to the gradient of the current that 00:29:54.310 --> 00:29:56.260 goes into that place. 00:29:56.260 --> 00:30:01.380 So the appropriate way of writing an equation that 00:30:01.380 --> 00:30:04.710 describes, let's say, the magnetization changing 00:30:04.710 --> 00:30:07.150 as a function of time. 00:30:07.150 --> 00:30:09.410 Given that you have a conservation, 00:30:09.410 --> 00:30:14.340 though, is to write it as minus the gradient 00:30:14.340 --> 00:30:15.670 of some kind of a current. 00:30:19.895 --> 00:30:22.980 So this j is some kind of a current, 00:30:22.980 --> 00:30:25.045 and these would be vectors. 00:30:29.770 --> 00:30:32.720 This is a current of the particles 00:30:32.720 --> 00:30:36.760 moving into the system. 00:30:36.760 --> 00:30:42.150 Now, in systems that are dissipative, 00:30:42.150 --> 00:30:46.260 currents are related to the gradient of some density 00:30:46.260 --> 00:30:49.210 through the diffusion constant, et cetera. 00:30:49.210 --> 00:30:56.360 So it kind of makes sense to imagine that this current is 00:30:56.360 --> 00:31:03.840 the gradient of something that is trying-- or more precisely, 00:31:03.840 --> 00:31:06.780 minus the gradient of something that 00:31:06.780 --> 00:31:10.790 tries to bring the system to be, more or less, 00:31:10.790 --> 00:31:13.330 in its equilibrium state. 00:31:13.330 --> 00:31:18.740 Equilibrium state, as we said, is determined by this data H, 00:31:18.740 --> 00:31:21.890 and we want to push it in that direction. 00:31:21.890 --> 00:31:29.250 So we put our data H by dm over here, 00:31:29.250 --> 00:31:32.283 and we put some kind of a U over here. 00:31:35.670 --> 00:31:39.370 Of course, I would have to add some kind 00:31:39.370 --> 00:31:43.180 of a conserved random current also, 00:31:43.180 --> 00:31:47.640 which is the analog of this non-conserved noise 00:31:47.640 --> 00:31:50.507 that I add over initially. 00:31:50.507 --> 00:31:51.007 OK. 00:31:55.350 --> 00:32:00.450 Now, the conservative version of the equation, you can see, 00:32:00.450 --> 00:32:04.870 you have two more derivatives with respect 00:32:04.870 --> 00:32:07.340 to what we had before. 00:32:07.340 --> 00:32:10.590 And so once I-- OK. 00:32:10.590 --> 00:32:19.530 So if we do something like this, dm by dt is mu C. 00:32:19.530 --> 00:32:22.000 And then I would have [INAUDIBLE] 00:32:22.000 --> 00:32:30.170 plus of R, rather than by itself. 00:32:30.170 --> 00:32:35.880 Actually, it would be the plus then of something like R m 00:32:35.880 --> 00:32:41.701 plus 4 U m cubed, and so forth. 00:32:41.701 --> 00:32:47.520 And we are going to ignore this kind of term. 00:32:47.520 --> 00:32:50.150 And then there would be high order terms 00:32:50.150 --> 00:32:53.735 that would show up minus k the fourth derivative, 00:32:53.735 --> 00:32:55.130 and so forth. 00:32:59.320 --> 00:33:04.380 And then there may be some kind of a conserved noise 00:33:04.380 --> 00:33:06.220 that I have to put outside. 00:33:11.055 --> 00:33:11.555 OK. 00:33:14.960 --> 00:33:18.890 So when I fully transform this equation, what do I get? 00:33:18.890 --> 00:33:27.530 I will get that dm by dt is-- let's say in the full space, 00:33:27.530 --> 00:33:34.520 until there is a function of qnt is minus 00:33:34.520 --> 00:33:39.770 U C. Because of this plus, then there's an additional factor 00:33:39.770 --> 00:33:42.040 of q squared. 00:33:42.040 --> 00:33:48.540 And then I have R plus kq squared plus L 00:33:48.540 --> 00:33:51.740 cubed to the fourth, et cetera. 00:33:51.740 --> 00:33:54.960 And then I will have a fully transformed version 00:33:54.960 --> 00:33:58.529 of this conserved noise. 00:33:58.529 --> 00:33:59.028 OK. 00:34:02.280 --> 00:34:08.060 You can see that the difference between this equation 00:34:08.060 --> 00:34:12.480 and the previous equation is that all of the relaxation 00:34:12.480 --> 00:34:18.639 times will have an additional factor of q squared. 00:34:18.639 --> 00:34:25.635 And so eventually, this shortest relaxation time actually 00:34:25.635 --> 00:34:29.790 will glow like the size of the system squared. 00:34:29.790 --> 00:34:31.522 Whereas previously, it was saturated 00:34:31.522 --> 00:34:34.620 at the correlation length. 00:34:34.620 --> 00:34:37.639 And because you will have this conservation, 00:34:37.639 --> 00:34:42.280 though, you have to rearrange a lot of particles 00:34:42.280 --> 00:34:44.580 keeping their numbers constant. 00:34:44.580 --> 00:34:49.015 You have a much harder time of relaxing the system. 00:34:49.015 --> 00:34:53.389 All of the relaxation times, as we see, 00:34:53.389 --> 00:34:57.426 grow correspondingly and become higher. 00:34:57.426 --> 00:34:58.310 OK. 00:34:58.310 --> 00:35:04.940 So indeed, for this class, one can show that z starts with 4, 00:35:04.940 --> 00:35:07.940 and then there will be corrections 00:35:07.940 --> 00:35:11.110 that would modify that. 00:35:11.110 --> 00:35:16.298 So the-- yes? 00:35:16.298 --> 00:35:19.376 AUDIENCE: How do we define or how do we 00:35:19.376 --> 00:35:22.322 do a realization of the conserved noise, conserved 00:35:22.322 --> 00:35:25.396 current-- conserved noise in the room? 00:35:25.396 --> 00:35:26.146 MEHRAN KARDAR: OK. 00:35:26.146 --> 00:35:29.700 AUDIENCE: So it has some kind of like correlation-- 00:35:29.700 --> 00:35:32.790 self-correlation properties, I suppose, 00:35:32.790 --> 00:35:36.080 because, if current flowing out of some region, 00:35:36.080 --> 00:35:39.430 doesn't it want to go in? 00:35:39.430 --> 00:35:41.600 MEHRAN KARDAR: If I go back here, 00:35:41.600 --> 00:35:46.270 I have a good idea of what is happening because all I need, 00:35:46.270 --> 00:35:50.930 in order to ensure conservation, is that the m by dt 00:35:50.930 --> 00:35:53.053 is the gradient of something. 00:35:53.053 --> 00:35:53.920 AUDIENCE: OK. 00:35:53.920 --> 00:35:58.015 MEHRAN KARDAR: So I can put whatever I want over here. 00:35:58.015 --> 00:36:00.619 AUDIENCE: So if it's a scalar or an [INAUDIBLE] field? 00:36:00.619 --> 00:36:01.410 MEHRAN KARDAR: Yes. 00:36:01.410 --> 00:36:04.019 As long as it is sitting under the gradient-- 00:36:04.019 --> 00:36:04.560 AUDIENCE: OK. 00:36:04.560 --> 00:36:07.610 MEHRAN KARDAR: --it will be OK, which means this quantity here 00:36:07.610 --> 00:36:12.730 that I'm calling a to z has a gradient in it. 00:36:12.730 --> 00:36:15.640 And if you wait for about five minutes, 00:36:15.640 --> 00:36:19.030 we'll show that, because of that in full space, 00:36:19.030 --> 00:36:22.590 rather than having-- well, I'll describe 00:36:22.590 --> 00:36:25.060 the difference between non-conserved and conserved 00:36:25.060 --> 00:36:26.770 noise in fully space. 00:36:26.770 --> 00:36:30.046 It's much easier. 00:36:30.046 --> 00:36:30.546 OK. 00:36:35.010 --> 00:36:40.750 So actually, as far as what I have discussed so far, which 00:36:40.750 --> 00:36:45.340 is relaxation, I don't really need the noise 00:36:45.340 --> 00:36:47.790 because I can forget the noise. 00:36:47.790 --> 00:36:49.360 And all I have said-- and I forgot 00:36:49.360 --> 00:36:54.170 the n tilde-- is that I have a linear equation that 00:36:54.170 --> 00:36:56.230 relaxes your variable to 0. 00:36:56.230 --> 00:36:59.175 I can immediately read off for the correlation, 00:36:59.175 --> 00:37:05.920 then this-- a correlation times what I need the noise for it 00:37:05.920 --> 00:37:11.410 so that, ultimately, I don't go to the medium is the potential, 00:37:11.410 --> 00:37:15.050 but I go to this pro-rated distribution. 00:37:15.050 --> 00:37:20.880 So let's see what we have to do in order to achieve that. 00:37:20.880 --> 00:37:24.235 For simplicity, let's take this equation. 00:37:28.380 --> 00:37:34.940 Although I can take the corresponding one for that. 00:37:34.940 --> 00:37:39.650 And let's calculate-- because of the presence of this noise, 00:37:39.650 --> 00:37:44.260 if I run the same system at different times, 00:37:44.260 --> 00:37:46.930 I will have different realizations of the noise 00:37:46.930 --> 00:37:49.930 than if I had run many versions of the system because 00:37:49.930 --> 00:37:52.020 of the realizations of noise. 00:37:52.020 --> 00:37:55.315 It's quantity and tilde would be different. 00:37:55.315 --> 00:37:59.540 It would satisfy some kind of a pro-rated distribution. 00:37:59.540 --> 00:38:02.480 So what I want to do is to calculate averages, 00:38:02.480 --> 00:38:05.280 such as the average of m tilde. 00:38:05.280 --> 00:38:11.920 Let's say, q1 at time T with m tilde q2 at time t. 00:38:15.340 --> 00:38:21.870 And you can see already from this equation 00:38:21.870 --> 00:38:26.250 that, if I forget the part that comes from the noise, 00:38:26.250 --> 00:38:28.755 whatever initial condition that I have 00:38:28.755 --> 00:38:31.440 will eventually decay to 0. 00:38:31.440 --> 00:38:33.640 So the thing that agitates and gives 00:38:33.640 --> 00:38:38.680 some kind of a randomness to this really comes from this. 00:38:38.680 --> 00:38:40.620 So let's imagine that we have looked 00:38:40.620 --> 00:38:43.620 at times that are sufficiently long so 00:38:43.620 --> 00:38:45.880 that the influence of the initial condition 00:38:45.880 --> 00:38:46.950 has died down. 00:38:46.950 --> 00:38:48.720 I don't want to write the other term. 00:38:48.720 --> 00:38:53.570 I could do it, but it's kind of boring to include it. 00:38:53.570 --> 00:38:57.340 So let's forget that and focus on the integral. 00:38:57.340 --> 00:39:00.140 0 to t. 00:39:00.140 --> 00:39:04.230 Now, if I multiply two of these quantities, 00:39:04.230 --> 00:39:08.318 I will have two integrals over t prime. 00:39:08.318 --> 00:39:10.290 All right. 00:39:10.290 --> 00:39:16.625 Each one of them would decay with the corresponding tau 00:39:16.625 --> 00:39:18.170 of q. 00:39:18.170 --> 00:39:20.260 In one case, tau of q1. 00:39:20.260 --> 00:39:27.080 In the other case, tau of q2. 00:39:27.080 --> 00:39:29.430 Coming from these things. 00:39:29.430 --> 00:39:35.780 And the noise, q1 at time to 1 prime, and noise q2 00:39:35.780 --> 00:39:39.764 at time [? t2 prime. ?] 00:39:39.764 --> 00:39:41.760 OK. 00:39:41.760 --> 00:39:48.050 Now if I average over the noise, then I 00:39:48.050 --> 00:39:51.414 have to do an average over here. 00:39:51.414 --> 00:39:51.913 OK. 00:39:55.140 --> 00:40:01.310 Now, one thing that I forgot to mention right at the beginning 00:40:01.310 --> 00:40:04.146 is that, of course, the average of this 00:40:04.146 --> 00:40:05.880 we are going to set to 0. 00:40:05.880 --> 00:40:08.020 It's the very least that is important. 00:40:08.020 --> 00:40:09.520 Right? 00:40:09.520 --> 00:40:11.550 So if I do that, clearly, the average 00:40:11.550 --> 00:40:14.280 of one of these in full space would 00:40:14.280 --> 00:40:21.950 be 0 also because the full q is related to the real space delta 00:40:21.950 --> 00:40:24.750 just by an integral. 00:40:24.750 --> 00:40:26.375 So if the average of the integral is 0, 00:40:26.375 --> 00:40:29.390 the average of this is 0. 00:40:29.390 --> 00:40:33.200 So it turns out that, when you look at the average of two 00:40:33.200 --> 00:40:36.010 of them-- and it's a very simple exercise 00:40:36.010 --> 00:40:43.115 to just rewrite these things in terms of 8 of x and t. 00:40:43.115 --> 00:40:47.490 8 of x and t applied average that you have. 00:40:47.490 --> 00:40:52.840 And we find that the things that are uncorrelated in real space 00:40:52.840 --> 00:40:56.280 also are uncoordinated in full space. 00:40:56.280 --> 00:40:58.970 And so the variance of this quantity 00:40:58.970 --> 00:41:05.000 is 2d delta 1 prime minus [? d2 prime. ?] 00:41:05.000 --> 00:41:09.950 And then you have the analog of the function in full space, 00:41:09.950 --> 00:41:16.330 which always carries the solution of a factor of 2 pi 00:41:16.330 --> 00:41:17.370 to the d. 00:41:17.370 --> 00:41:20.000 And it becomes a sum of the q's, as we've 00:41:20.000 --> 00:41:22.860 seen many times before. 00:41:22.860 --> 00:41:25.060 OK. 00:41:25.060 --> 00:41:28.350 So because of this delta function, 00:41:28.350 --> 00:41:30.920 this delta integral becomes one integral. 00:41:30.920 --> 00:41:34.480 So I have the integral 0 to t. 00:41:34.480 --> 00:41:39.170 The 2t prime I can write as just 1t prime, 00:41:39.170 --> 00:41:42.990 and these two factors merge into one. 00:41:42.990 --> 00:41:49.680 It comes into minus t minus t prime over tau of q, 00:41:49.680 --> 00:41:53.650 except that I get multiplied by a factor 2 00:41:53.650 --> 00:41:56.900 since I have two of them. 00:41:56.900 --> 00:42:05.640 And outside of the integral, I will have this factor of 2d, 00:42:05.640 --> 00:42:10.875 and then 2 pi to the d delta function with 1 plus 2. 00:42:15.125 --> 00:42:15.625 OK. 00:42:19.430 --> 00:42:21.880 Now, we are really interested-- and I already 00:42:21.880 --> 00:42:24.825 kind of hinted at that in the limit 00:42:24.825 --> 00:42:29.130 where time becomes very large. 00:42:29.130 --> 00:42:32.930 In the limit, where time becomes a very large, 00:42:32.930 --> 00:42:36.630 essentially, I need to calculate the limit of this integral 00:42:36.630 --> 00:42:38.308 as time becomes very large. 00:42:42.132 --> 00:42:44.680 And as time grows to very large, this 00:42:44.680 --> 00:42:47.470 is just the integral from 0 to infinity. 00:42:47.470 --> 00:42:55.280 And the integral is going to give me 2 over tau of q. 00:42:55.280 --> 00:42:59.255 Essentially, you can see that integrating at the upper end 00:42:59.255 --> 00:43:02.810 will give me just 0. 00:43:02.810 --> 00:43:05.445 Integrating at the smaller end, it 00:43:05.445 --> 00:43:07.120 will be exponentially small as it 00:43:07.120 --> 00:43:11.855 goes to infinity as a factor of 2 over tau. 00:43:11.855 --> 00:43:12.750 OK. 00:43:12.750 --> 00:43:14.202 Yes? 00:43:14.202 --> 00:43:18.460 AUDIENCE: Are you assuming or-- yeah. 00:43:18.460 --> 00:43:22.230 Are you assuming that tau of q is even in q 00:43:22.230 --> 00:43:24.039 to be able to combine to-- 00:43:24.039 --> 00:43:24.830 MEHRAN KARDAR: Yes. 00:43:24.830 --> 00:43:25.320 AUDIENCE: --2 tau? 00:43:25.320 --> 00:43:25.850 So-- OK. 00:43:25.850 --> 00:43:28.183 MEHRAN KARDAR: I'm thinking of the tau of q's that we've 00:43:28.183 --> 00:43:29.201 calculated over here. 00:43:29.201 --> 00:43:29.742 AUDIENCE: OK. 00:43:29.742 --> 00:43:30.990 MEHRAN KARDAR: OK. 00:43:30.990 --> 00:43:31.820 Yes? 00:43:31.820 --> 00:43:34.850 AUDIENCE: The e times the [INAUDIBLE] 00:43:34.850 --> 00:43:36.370 do the equation of something? 00:43:36.370 --> 00:43:37.980 Right? 00:43:37.980 --> 00:43:42.030 MEHRAN KARDAR: At this stage, I am focusing on this expression 00:43:42.030 --> 00:43:44.610 over here, where a is not. 00:43:44.610 --> 00:43:46.640 But I will come to that expression also. 00:43:49.900 --> 00:43:54.440 So for the time being, this d is just the same constant 00:43:54.440 --> 00:43:56.455 as we have over here. 00:43:56.455 --> 00:43:58.200 OK? 00:43:58.200 --> 00:44:01.310 So you can see that the final answer 00:44:01.310 --> 00:44:08.680 is going to be d over tau of q 2 pi 00:44:08.680 --> 00:44:12.314 to the d delta function q1 plus q2. 00:44:15.150 --> 00:44:21.700 And if I use the value of tau of q that I have, tau of q 00:44:21.700 --> 00:44:29.820 is mu R plus kq squared, which becomes d over mu R 00:44:29.820 --> 00:44:33.600 plus kq squared and so forth. 00:44:33.600 --> 00:44:38.310 2 pi to the d delta function q1 plus q2. 00:44:42.340 --> 00:44:49.510 So essentially, if I take this linearized time dependent line 00:44:49.510 --> 00:44:54.180 of Landau-Ginzburg equation, run it for a very long time, 00:44:54.180 --> 00:44:56.970 and look at the correlations of the field, 00:44:56.970 --> 00:44:59.340 I see that the correlations of the field, 00:44:59.340 --> 00:45:03.380 at the limit of long times, satisfy this expression. 00:45:06.220 --> 00:45:10.090 Now, what do I know if I look at the top line 00:45:10.090 --> 00:45:14.380 that I have for the probability of distribution? 00:45:14.380 --> 00:45:17.890 I can go and express that probability of distribution 00:45:17.890 --> 00:45:19.175 in full mode. 00:45:19.175 --> 00:45:22.160 In the linear version, I immediately 00:45:22.160 --> 00:45:27.580 get that the probability of m tilde of q 00:45:27.580 --> 00:45:31.960 is proportional to a product over different q's, 00:45:31.960 --> 00:45:36.890 e to the minus R plus kq squared, 00:45:36.890 --> 00:45:42.250 et cetera, and tilde of q squared over 2. 00:45:42.250 --> 00:45:44.600 All right. 00:45:44.600 --> 00:45:48.060 So when I look at the equilibrium linear 00:45:48.060 --> 00:45:51.730 as Landau-Ginzburg, I can see that, if I calculate 00:45:51.730 --> 00:45:57.560 the average of m of q1, m of q2, then 00:45:57.560 --> 00:46:01.950 this is an equilibrium average. 00:46:01.950 --> 00:46:06.980 What I would get is 2 pi to the d delta function 00:46:06.980 --> 00:46:10.870 of q1 plus q2 because, clearly, the different q's are recovered 00:46:10.870 --> 00:46:12.620 from each other. 00:46:12.620 --> 00:46:17.530 And for the particular value of q, what I will get is 1 over R 00:46:17.530 --> 00:46:21.030 plus kq squared and so forth. 00:46:21.030 --> 00:46:21.890 Yes? 00:46:21.890 --> 00:46:26.235 AUDIENCE: So isn't the 2 over tau-- [INAUDIBLE]? 00:46:26.235 --> 00:46:27.026 MEHRAN KARDAR: Yes. 00:46:36.132 --> 00:46:37.124 Over 2. 00:46:43.072 --> 00:46:43.572 Yeah. 00:46:54.976 --> 00:46:55.476 OK. 00:47:02.420 --> 00:47:03.412 OK. 00:47:03.412 --> 00:47:07.975 I changed because I had 2d cancels the 2. 00:47:07.975 --> 00:47:10.600 I would have to put here tau of q over 2. 00:47:10.600 --> 00:47:13.360 I had 2d times tau of q over 2. 00:47:13.360 --> 00:47:15.100 So it's d tau. 00:47:15.100 --> 00:47:18.618 And inverse of tau, I have everything correct. 00:47:18.618 --> 00:47:21.330 OK. 00:47:21.330 --> 00:47:28.031 So if you make an even number of errors, the answer comes up. 00:47:28.031 --> 00:47:28.530 OK. 00:47:28.530 --> 00:47:31.700 But you can now compare this expression 00:47:31.700 --> 00:47:36.940 that comes from equilibrium and this expression that 00:47:36.940 --> 00:47:42.760 comes from the long time limit of this noisy equation. 00:47:42.760 --> 00:47:48.120 OK So we want to choose our noise 00:47:48.120 --> 00:47:55.590 so that the stochastic dynamics gives the same value 00:47:55.590 --> 00:48:01.100 as equilibrium, just like we did for the case of a Brownian 00:48:01.100 --> 00:48:06.340 particular where you have some kind of an Einstein equation 00:48:06.340 --> 00:48:15.125 that was relating the strength of the noise and the mobility. 00:48:15.125 --> 00:48:19.120 And we see that here all I need to do 00:48:19.120 --> 00:48:27.198 is to ensure that d over mu should be equal to 1. 00:48:27.198 --> 00:48:27.698 OK. 00:48:33.011 --> 00:48:41.810 Now, the thing is that, if I am doing this, 00:48:41.810 --> 00:48:48.430 I, in principle, can have a different noise for each q, 00:48:48.430 --> 00:48:52.110 and compensate by different mobility for each q. 00:48:52.110 --> 00:48:55.350 And I would get the same answer. 00:48:55.350 --> 00:49:01.690 So in the non-conserved version of this time dependent dynamics 00:49:01.690 --> 00:49:07.300 that you wrote down, the d was a constant 00:49:07.300 --> 00:49:11.280 and the mu was a constant. 00:49:11.280 --> 00:49:14.940 Whereas, if you want to get the same equilibrium result out 00:49:14.940 --> 00:49:19.300 of the conserved dynamics, you can see that, essentially, 00:49:19.300 --> 00:49:23.650 what we previously had as mu became something that 00:49:23.650 --> 00:49:26.540 is proportional to q squared. 00:49:26.540 --> 00:49:31.540 So essentially, here, this becomes mu C q squared. 00:49:31.540 --> 00:49:35.070 So clearly, in order to get the same answer, 00:49:35.070 --> 00:49:42.750 I have to put my noise to be proportional to q squared also. 00:49:42.750 --> 00:49:46.570 And we can see that this kind of conserved noise that I put over 00:49:46.570 --> 00:49:49.990 here achieves that because, as I said, 00:49:49.990 --> 00:49:54.870 this conserved noise is the gradient of something, which 00:49:54.870 --> 00:49:58.090 means that, when I go to fully space, if it be q, 00:49:58.090 --> 00:50:00.740 it will be proportional to q. 00:50:00.740 --> 00:50:02.620 And when I take its variants, it's variants 00:50:02.620 --> 00:50:05.800 will be proportional to q squared. 00:50:05.800 --> 00:50:09.580 Anything precisely canceled. 00:50:09.580 --> 00:50:12.580 But you can see that you also-- this 00:50:12.580 --> 00:50:16.390 had a physical explanation in terms of a conservation law. 00:50:16.390 --> 00:50:20.450 In principle, you can cook up all kinds of b of q and mu 00:50:20.450 --> 00:50:22.070 of q. 00:50:22.070 --> 00:50:26.780 As long as this equality is satisfied, you will have, 00:50:26.780 --> 00:50:31.010 for these linear stochastic equations, 00:50:31.010 --> 00:50:35.381 the guarantee that you would always get the same equilibrium 00:50:35.381 --> 00:50:35.880 result. 00:50:35.880 --> 00:50:38.700 Because if you wait for this dynamics 00:50:38.700 --> 00:50:43.060 to settle down after long times, you will get to the answer. 00:50:43.060 --> 00:50:43.842 Yes? 00:50:43.842 --> 00:50:48.312 AUDIENCE: I wonder how general is this result for stochastic 00:50:48.312 --> 00:50:50.070 [INAUDIBLE]? 00:50:50.070 --> 00:50:50.820 MEHRAN KARDAR: OK. 00:50:50.820 --> 00:50:51.736 AUDIENCE: But what I-- 00:50:51.736 --> 00:50:54.380 MEHRAN KARDAR: So what I showed you 00:50:54.380 --> 00:50:58.570 was with for linearized version, and the only thing that I 00:50:58.570 --> 00:51:01.220 calculated was the variance. 00:51:01.220 --> 00:51:04.090 And I showed that the variances were the same. 00:51:04.090 --> 00:51:06.310 And if I have a Gaussian problem of distribution, 00:51:06.310 --> 00:51:08.920 the variance is completely categorizable 00:51:08.920 --> 00:51:10.360 with distribution. 00:51:10.360 --> 00:51:11.880 So this is safe. 00:51:11.880 --> 00:51:15.580 But we are truly interested in the more general 00:51:15.580 --> 00:51:18.740 non-Gaussian probability of distribution. 00:51:18.740 --> 00:51:24.440 So the question really is-- if I keep the full non-linearity 00:51:24.440 --> 00:51:29.730 in this story, would I be able to show 00:51:29.730 --> 00:51:31.830 that the probability of distribution that 00:51:31.830 --> 00:51:36.140 will be characterized by all kinds of moments 00:51:36.140 --> 00:51:38.472 eventually has the same behavior as that. 00:51:38.472 --> 00:51:39.180 AUDIENCE: Mm-hmm. 00:51:39.180 --> 00:51:41.460 MEHRAN KARDAR: And the answer is, in fact, yes. 00:51:41.460 --> 00:51:46.310 There's a procedure that relies on converting 00:51:46.310 --> 00:51:48.940 this equation-- sorry. 00:51:48.940 --> 00:51:51.660 One equation that governs the evolution 00:51:51.660 --> 00:51:55.421 of the full probability as a function of time. 00:51:55.421 --> 00:51:55.920 Right? 00:51:55.920 --> 00:52:00.930 So basically, I can start with a an initial probability 00:52:00.930 --> 00:52:04.430 and see how this probability evolves as a function of time. 00:52:04.430 --> 00:52:08.220 And this is sometimes called a master equation. 00:52:08.220 --> 00:52:10.255 Sometimes, called a [INAUDIBLE] equation. 00:52:13.000 --> 00:52:20.790 And we did cover, in fact, this in-- next spring 00:52:20.790 --> 00:52:23.370 in the statistical physics and biology, 00:52:23.370 --> 00:52:25.630 we spent some time talking about these things. 00:52:25.630 --> 00:52:29.390 So you can come back to the third version of this class. 00:52:29.390 --> 00:52:35.015 And one can ensure that, with appropriate choice 00:52:35.015 --> 00:52:40.270 of the noise, the asymptotic solution for this probability 00:52:40.270 --> 00:52:43.880 distribution is whatever Landau-Ginzburg 00:52:43.880 --> 00:52:45.600 or other probability distribution 00:52:45.600 --> 00:52:47.458 that you need most. 00:52:47.458 --> 00:52:51.130 AUDIENCE: So is this true if we assume 00:52:51.130 --> 00:52:54.140 Landau-Ginzburg potential for how resistant? 00:52:54.140 --> 00:52:54.975 MEHRAN KARDAR: Yes. 00:52:54.975 --> 00:52:55.965 AUDIENCE: OK. 00:52:55.965 --> 00:52:59.200 Maybe this is not a very good-stated question, 00:52:59.200 --> 00:53:01.705 but is there kind of like an even more general level? 00:53:01.705 --> 00:53:03.330 MEHRAN KARDAR: I'll come to that, sure. 00:53:03.330 --> 00:53:06.390 But currently, the way that I set up the problem 00:53:06.390 --> 00:53:09.110 was that we know some complicated equilibrium 00:53:09.110 --> 00:53:13.250 force that exist-- form of the probability that exist. 00:53:13.250 --> 00:53:16.510 And these kinds of linear-- these kinds 00:53:16.510 --> 00:53:20.090 of stochastic linear or non-linear evolution 00:53:20.090 --> 00:53:22.340 equations-- generally called non- [INAUDIBLE] 00:53:22.340 --> 00:53:26.150 equations-- one can show that, with the appropriate choice 00:53:26.150 --> 00:53:30.690 of the noise, we'll be able to asymptotically reproduce 00:53:30.690 --> 00:53:34.192 the probability of distribution that we knew. 00:53:34.192 --> 00:53:35.894 But now, the question is, of course, you 00:53:35.894 --> 00:53:37.685 don't know the probability of distribution. 00:53:37.685 --> 00:53:40.113 And I'll say a few words about that. 00:53:40.113 --> 00:53:40.654 AUDIENCE: OK. 00:53:40.654 --> 00:53:41.570 Thank you. 00:53:41.570 --> 00:53:44.540 MEHRAN KARDAR: Anything else? 00:53:44.540 --> 00:53:45.420 OK. 00:53:45.420 --> 00:53:52.130 So the lesson of this part is that the field 00:53:52.130 --> 00:53:58.070 of dynamic or critical phenomena is quite rich, much richer 00:53:58.070 --> 00:54:01.450 than the corresponding equilibrium critical phenomena 00:54:01.450 --> 00:54:04.610 because the same equilibrium state can 00:54:04.610 --> 00:54:08.360 be obtained by various different types of dynamics. 00:54:08.360 --> 00:54:12.400 And I explained to you just one conservation law, 00:54:12.400 --> 00:54:14.090 but there could be some combination 00:54:14.090 --> 00:54:17.020 of conservation of energy, conservation of something. 00:54:17.020 --> 00:54:21.360 So there is a whole listing of different universality classes 00:54:21.360 --> 00:54:26.254 that people have targeted for the dynamics. 00:54:26.254 --> 00:54:30.620 But not all of this was assuming that you 00:54:30.620 --> 00:54:35.800 know what the ultimate answer is because, in all cases, 00:54:35.800 --> 00:54:38.870 the equations that we're writing are 00:54:38.870 --> 00:54:43.910 dependent on some kind of a gradient descent conserved 00:54:43.910 --> 00:54:47.100 or non-conserved around something that corresponded 00:54:47.100 --> 00:54:49.460 to the log of probability of distribution 00:54:49.460 --> 00:54:52.190 that we eventually want to put. 00:54:52.190 --> 00:54:55.770 And maybe you don't know that, and so 00:54:55.770 --> 00:55:04.000 let me give you a particular example in the context of, 00:55:04.000 --> 00:55:06.464 let's say, surface interface fluctuations. 00:55:12.160 --> 00:55:15.160 Starting from things that you know and then building 00:55:15.160 --> 00:55:18.780 to something that maybe you don't. 00:55:18.780 --> 00:55:21.888 Let's first with the case of a soap bubble. 00:55:25.720 --> 00:55:30.680 So we take some kind of a circle or whatever, 00:55:30.680 --> 00:55:34.980 and we put a soap bubble on top of it. 00:55:34.980 --> 00:55:41.080 And in this case, the energy of the formation-- 00:55:41.080 --> 00:55:43.780 the cost of the formation comes from surface tension. 00:55:50.020 --> 00:55:54.900 And let's say, the cost of the deformation 00:55:54.900 --> 00:55:58.830 is the changing area times some sigma. 00:56:04.450 --> 00:56:10.160 So I neglect the contribution that 00:56:10.160 --> 00:56:13.870 comes from the flat surface, and see if I make a deformation. 00:56:13.870 --> 00:56:17.460 If I make a deformation, I have changed the area of the spin. 00:56:17.460 --> 00:56:20.220 So there is a cost that is proportion to the surface 00:56:20.220 --> 00:56:22.930 tension times the change in area. 00:56:22.930 --> 00:56:26.230 Change in area locally is the square root of 1 00:56:26.230 --> 00:56:29.240 plus the gradient of a height profile. 00:56:29.240 --> 00:56:34.750 So what I can do is I can define at each point on the surface 00:56:34.750 --> 00:56:39.420 how much it has changed its height from being perfectly 00:56:39.420 --> 00:56:39.920 flat. 00:56:39.920 --> 00:56:42.320 So h equals to 0 is flat. 00:56:42.320 --> 00:56:47.770 Local area is the integral dx dy of square root of 1 00:56:47.770 --> 00:56:50.790 plus gradient of h squared, minus 1. 00:56:50.790 --> 00:56:53.060 That corresponds to the flat. 00:56:53.060 --> 00:56:54.420 And so then you expand that. 00:56:54.420 --> 00:56:58.360 The first term is going to be the integral gradient of h 00:56:58.360 --> 00:56:58.860 squared. 00:57:01.500 --> 00:57:06.490 So this is the analog of what we had over there, only 00:57:06.490 --> 00:57:07.640 the first term. 00:57:07.640 --> 00:57:10.440 So you would say that the equation that you would write 00:57:10.440 --> 00:57:16.210 down for this would be all to some constant mu 00:57:16.210 --> 00:57:18.690 proportional to the variations of this, 00:57:18.690 --> 00:57:24.260 which will give me something like sigma Laplacian of h. 00:57:24.260 --> 00:57:27.970 But because of the particles from the air constantly 00:57:27.970 --> 00:57:30.380 bombarding the surface, there will 00:57:30.380 --> 00:57:32.897 be some noise that depends on where 00:57:32.897 --> 00:57:36.108 you are on the surface in time. 00:57:36.108 --> 00:57:40.335 And this is the non-conserved version. 00:57:44.810 --> 00:57:48.030 And you can from this very quickly get 00:57:48.030 --> 00:57:53.610 that the expectation value of h tilde of q squared 00:57:53.610 --> 00:57:58.730 is going to be proportional to something like D 00:57:58.730 --> 00:58:03.840 over mu sigma q squared, because of this q squared. 00:58:03.840 --> 00:58:06.990 And if you ask how much fluctuations you 00:58:06.990 --> 00:58:10.170 have in real space-- so that typical scale 00:58:10.170 --> 00:58:13.620 of the fluctuations in real space-- 00:58:13.620 --> 00:58:18.010 will come from integrating 1 over q squared. 00:58:18.010 --> 00:58:21.110 And it's going to be our usual things that 00:58:21.110 --> 00:58:23.330 have this logarithmic dependence, so there will 00:58:23.330 --> 00:58:26.850 be something that ultimately will go logarithmically 00:58:26.850 --> 00:58:28.960 with the size of the system. 00:58:28.960 --> 00:58:31.440 The constant of proportionality will be 00:58:31.440 --> 00:58:34.080 proportional to kt over sigma. 00:58:34.080 --> 00:58:38.900 So you have to choose your D and mu to correspond to this. 00:58:38.900 --> 00:58:42.750 But basically, a soap film, as an example 00:58:42.750 --> 00:58:45.970 of all kinds of Goldstone mode-like things 00:58:45.970 --> 00:58:47.410 that we have seen. 00:58:47.410 --> 00:58:49.890 It's a 2-dimensional entity. 00:58:49.890 --> 00:58:53.120 We will have logarithmic fluctuations-- not very big, 00:58:53.120 --> 00:58:56.560 but ultimately, at large enough distances, 00:58:56.560 --> 00:58:59.110 it will have fluctuations. 00:58:59.110 --> 00:59:02.230 So that was non-conserved. 00:59:02.230 --> 00:59:04.750 I can imagine that, rather than this, 00:59:04.750 --> 00:59:12.150 I have the case of a surface of a pool. 00:59:12.150 --> 00:59:17.890 So here I have some depth of water, 00:59:17.890 --> 00:59:21.930 and then there's the surface of the pool of water. 00:59:21.930 --> 00:59:26.730 And the difference between this case and the previous case-- 00:59:26.730 --> 00:59:29.530 both of them can be described by a height function. 00:59:33.130 --> 00:59:36.280 The difference is that if I ignore evaporation 00:59:36.280 --> 00:59:41.020 and condensation, the total mass of water 00:59:41.020 --> 00:59:42.800 is going to be conserved. 00:59:42.800 --> 00:59:49.420 So I would need to have divided t of the integral dx d qx 00:59:49.420 --> 00:59:55.060 h of x and t to be 0. 00:59:55.060 --> 00:59:57.746 So this would go into the conserved variety. 01:00:00.500 --> 01:00:05.120 And while, if I create a ripple on the surface of this 01:00:05.120 --> 01:00:10.770 compared to the surface of that, the relaxation time 01:00:10.770 --> 01:00:13.320 through this dissipative dynamics 01:00:13.320 --> 01:00:17.280 would be much longer in this case as opposed to that case. 01:00:17.280 --> 01:00:20.390 Ultimately, if I wait sufficiently long time, 01:00:20.390 --> 01:00:24.640 both of them would have exactly the same fluctuations. 01:00:24.640 --> 01:00:27.710 That is, you would go logarithmically with the length 01:00:27.710 --> 01:00:31.050 scale over which [INAUDIBLE]. 01:00:31.050 --> 01:00:37.170 OK, so now let's look at another system that fluctuates. 01:00:37.170 --> 01:00:40.620 And I don't know what the final answer is. 01:00:40.620 --> 01:00:43.240 That was the question, maybe, that you asked. 01:00:43.240 --> 01:00:46.620 The example that I will give is the following-- 01:00:46.620 --> 01:00:50.480 so suppose that you have a surface. 01:00:50.480 --> 01:00:56.330 And then you have a rain of sticky materials 01:00:56.330 --> 01:00:59.670 that falls down on top of it. 01:00:59.670 --> 01:01:03.500 So this material will come down. 01:01:03.500 --> 01:01:05.400 You'll have something like this. 01:01:05.400 --> 01:01:07.860 And then as time goes on, there will 01:01:07.860 --> 01:01:12.090 be more material that will come, more material that will come, 01:01:12.090 --> 01:01:14.210 more material that will come. 01:01:14.210 --> 01:01:20.840 So there, because the particles are raining down randomly 01:01:20.840 --> 01:01:25.810 at different points, there will be a stochastic process 01:01:25.810 --> 01:01:27.450 that is going on. 01:01:27.450 --> 01:01:32.230 So you can try to characterize the system 01:01:32.230 --> 01:01:35.360 in terms of a height that changes as a function of t 01:01:35.360 --> 01:01:38.890 and as a function of position. 01:01:38.890 --> 01:01:44.930 And there could be all kinds of microscopic things going on, 01:01:44.930 --> 01:01:46.930 like maybe these are particles that 01:01:46.930 --> 01:01:51.260 are representing some kind of a deposition process. 01:01:51.260 --> 01:01:54.390 And then they come, they stick in a particular way. 01:01:54.390 --> 01:01:56.580 Maybe they can slide on the surface. 01:01:56.580 --> 01:02:01.360 We can imagine all kinds of microscopic degrees of freedom 01:02:01.360 --> 01:02:04.200 and things that we can put. 01:02:04.200 --> 01:02:08.660 But you say, well, can I change my perspective, 01:02:08.660 --> 01:02:12.500 and try to describe the system the same way that we 01:02:12.500 --> 01:02:16.180 did for the case of coarse grading and going 01:02:16.180 --> 01:02:18.010 from without the microscopic details 01:02:18.010 --> 01:02:22.930 to describe the phenomenological Landau-Ginzburg equation? 01:02:22.930 --> 01:02:26.910 And so you say, OK, there is a height that is growing. 01:02:26.910 --> 01:02:30.660 And what I will write down is an equation 01:02:30.660 --> 01:02:32.510 that is very similar to the equations 01:02:32.510 --> 01:02:36.090 that I had written before. 01:02:36.090 --> 01:02:40.340 Now I'm going to follow the same kind of reasoning 01:02:40.340 --> 01:02:44.570 that we did in the construction of this Landau-Ginzburg model, 01:02:44.570 --> 01:02:49.350 is we said that this weight is going to depend on all 01:02:49.350 --> 01:02:52.660 kinds of things that relate to this height 01:02:52.660 --> 01:02:53.870 that I don't quite know. 01:02:53.870 --> 01:02:58.230 So let's imagine that there is some kind of a function 01:02:58.230 --> 01:03:01.570 of the height itself. 01:03:01.570 --> 01:03:04.420 And potentially, just like we did over there, 01:03:04.420 --> 01:03:08.176 the gradient of the height, five derivatives of the height, 01:03:08.176 --> 01:03:08.675 et cetera. 01:03:12.460 --> 01:03:16.280 And then I will start to make an expansion of this 01:03:16.280 --> 01:03:20.750 in the same spirit that I did for the Landau-Ginzburg Model, 01:03:20.750 --> 01:03:23.540 except that when I was doing the Landau-Ginzburg Model, 01:03:23.540 --> 01:03:26.840 I was doing the expansion at the level of looking 01:03:26.840 --> 01:03:30.450 at the probability distribution and the log of the probability. 01:03:30.450 --> 01:03:34.625 Here I'm making the expansion at the level of an equation that 01:03:34.625 --> 01:03:35.500 governs the dynamics. 01:03:38.580 --> 01:03:41.250 Of course, in this particular system, 01:03:41.250 --> 01:03:44.620 that's not the end of story, because the change in height 01:03:44.620 --> 01:03:49.600 is also governed by this random addition of the particles. 01:03:49.600 --> 01:03:53.080 So there is some function that changes 01:03:53.080 --> 01:03:55.070 as a function of position and time, 01:03:55.070 --> 01:03:57.040 depending on whether, at that time, 01:03:57.040 --> 01:03:59.370 a particle was dropped down. 01:03:59.370 --> 01:04:01.920 I can always take the average of this 01:04:01.920 --> 01:04:07.740 to be 0, and put that average into the expansion of this 01:04:07.740 --> 01:04:10.220 starting from a constant. 01:04:10.220 --> 01:04:13.200 Basically, if I just have a single point 01:04:13.200 --> 01:04:17.010 and I randomly drop particles at that single point, 01:04:17.010 --> 01:04:19.560 there will be an average growth velocity, 01:04:19.560 --> 01:04:21.850 an average addition to the height, 01:04:21.850 --> 01:04:23.530 that averages over here. 01:04:23.530 --> 01:04:27.580 But there will be fluctuations that are going [INAUDIBLE]. 01:04:27.580 --> 01:04:31.360 OK but the constant is the first term 01:04:31.360 --> 01:04:34.330 in an expansion such as this. 01:04:34.330 --> 01:04:38.220 And you can start thinking, OK, what next order? 01:04:38.220 --> 01:04:41.300 Can I put something like alpha h? 01:04:41.300 --> 01:04:43.840 Potentially-- depends on your system-- 01:04:43.840 --> 01:04:49.440 but if the system is invariant whether you started from here 01:04:49.440 --> 01:04:52.490 or whether you started from there-- something like gravity, 01:04:52.490 --> 01:04:55.540 for example, is not important-- you say, OK, 01:04:55.540 --> 01:05:00.110 I cannot have any function of h if my dynamics will proceed 01:05:00.110 --> 01:05:05.330 exactly the same way if I were to translate this surface 01:05:05.330 --> 01:05:07.590 to some further up or further down. 01:05:07.590 --> 01:05:10.760 If I see that there's no change in future dynamics on average, 01:05:10.760 --> 01:05:14.660 then the dynamic cannot depend on this. 01:05:14.660 --> 01:05:16.970 OK, so we've got rid of that. 01:05:16.970 --> 01:05:19.370 And any function of h-- can I put 01:05:19.370 --> 01:05:23.640 something that is proportional to gradient of h? 01:05:23.640 --> 01:05:28.070 Maybe for something I can, but for h itself I cannot, 01:05:28.070 --> 01:05:30.510 because h is a scalar gradient. 01:05:30.510 --> 01:05:33.990 If h is a vector, I can't set something 01:05:33.990 --> 01:05:40.620 that is a scalar equal to a vector, so I can't have this. 01:05:40.620 --> 01:05:41.914 Yes? 01:05:41.914 --> 01:05:43.455 AUDIENCE: Couldn't you, in principle, 01:05:43.455 --> 01:05:47.130 make your constant term in front of the gradient also 01:05:47.130 --> 01:05:48.570 a vector and [INAUDIBLE]? 01:05:50.659 --> 01:05:51.700 MEHRAN KARDAR: You could. 01:05:51.700 --> 01:05:54.690 So there's a whole set of different systems 01:05:54.690 --> 01:05:56.630 that you can be thinking about. 01:05:56.630 --> 01:06:00.220 Right now, I want to focus on the simplest system, which 01:06:00.220 --> 01:06:02.770 is a scalar field, so that my equation can 01:06:02.770 --> 01:06:05.404 be as simple as possible, but still we 01:06:05.404 --> 01:06:07.070 will see it has sufficient complication. 01:06:10.640 --> 01:06:12.940 So you can see that if I don't have them, 01:06:12.940 --> 01:06:15.770 the next order term that I can have 01:06:15.770 --> 01:06:17.480 would be something like a Laplacian. 01:06:20.560 --> 01:06:24.440 So this kind of diffusion equation, you can see, 01:06:24.440 --> 01:06:28.780 has to emerge as a low-order expansion of something 01:06:28.780 --> 01:06:29.280 like this. 01:06:29.280 --> 01:06:32.140 And this is the ubiquity of the diffusion equation appearing 01:06:32.140 --> 01:06:34.380 all over the place. 01:06:34.380 --> 01:06:36.220 And then you could have terms that 01:06:36.220 --> 01:06:38.890 would be of the order of the fourth derivative, 01:06:38.890 --> 01:06:40.520 and so forth. 01:06:40.520 --> 01:06:42.480 There's nothing wrong with that. 01:06:46.410 --> 01:06:49.940 And then, if you think about it, you'll 01:06:49.940 --> 01:06:55.420 see that there is one interesting possibility that 01:06:55.420 --> 01:06:58.980 is not allowed for that system, but is 01:06:58.980 --> 01:07:03.950 allowed for this system, which is something that is a scalar. 01:07:03.950 --> 01:07:06.366 It's the gradient of h squared. 01:07:09.320 --> 01:07:15.100 Now I could not have added this term for the case of the soap 01:07:15.100 --> 01:07:17.960 bubble for the following reason-- 01:07:17.960 --> 01:07:22.760 that if I reverse the soap bubble so that h 01:07:22.760 --> 01:07:27.050 becomes minus h, the dynamics would proceed exactly 01:07:27.050 --> 01:07:28.960 as before. 01:07:28.960 --> 01:07:34.020 So the soap bubble has a symmetry of h going to minus h, 01:07:34.020 --> 01:07:38.160 and so that symmetry should be preserved in the equation. 01:07:38.160 --> 01:07:41.820 This term breaks that symmetry because the left-hand side 01:07:41.820 --> 01:07:45.760 is odd in h, whereas the right-hand side of this term 01:07:45.760 --> 01:07:48.080 would be even in h. 01:07:48.080 --> 01:07:51.990 But for the case of the growing surface-- and you've seen 01:07:51.990 --> 01:07:53.140 things that are growing. 01:07:53.140 --> 01:07:55.520 And typically, if I give you something 01:07:55.520 --> 01:07:59.270 that has grown like the tree trunk, for example, 01:07:59.270 --> 01:08:02.400 and if I take the picture of a part of it, 01:08:02.400 --> 01:08:05.370 and you don't see where the center is, where the end is, 01:08:05.370 --> 01:08:08.040 you can immediately tell from the way 01:08:08.040 --> 01:08:11.760 that the shape of this object is, that it is growing 01:08:11.760 --> 01:08:13.960 in some particular direction. 01:08:13.960 --> 01:08:17.590 So for growth systems, that symmetry does not exist. 01:08:17.590 --> 01:08:22.220 You are allowed to have this term, and so forth. 01:08:22.220 --> 01:08:26.300 Now the interesting thing about this term 01:08:26.300 --> 01:08:31.465 is that there is no beta h that you can write down 01:08:31.465 --> 01:08:36.040 that is local-- some function of h such 01:08:36.040 --> 01:08:40.229 that if you take a functional derivative with respect to h, 01:08:40.229 --> 01:08:46.310 it will reproduce that term-- just does not exist. 01:08:46.310 --> 01:08:51.750 So you can see that somehow immediately, 01:08:51.750 --> 01:08:59.380 as soon as we liberate ourselves from writing equations that 01:08:59.380 --> 01:09:03.899 came from functional derivative of something, 01:09:03.899 --> 01:09:07.310 but potentially have physical significance, 01:09:07.310 --> 01:09:10.810 we can write down new terms. 01:09:10.810 --> 01:09:14.000 So this is actually-- also, you can do 01:09:14.000 --> 01:09:17.020 this, even for two particles. 01:09:17.020 --> 01:09:22.770 A potential v of x1 and x2 will have some kind of derivatives. 01:09:22.770 --> 01:09:24.505 But if you write dynamical equations, 01:09:24.505 --> 01:09:26.600 there are dynamical equations that 01:09:26.600 --> 01:09:31.080 allow you to rotate 1 x from x1 to x2. 01:09:31.080 --> 01:09:32.759 That kind of term will never come 01:09:32.759 --> 01:09:34.369 from taking the derivative. 01:09:37.130 --> 01:09:38.330 So fine. 01:09:41.370 --> 01:09:43.490 So this is a candidate equation that 01:09:43.490 --> 01:09:47.420 is obtained in this context-- something that is grown. 01:09:47.420 --> 01:09:49.380 We say we are not interested in it's 01:09:49.380 --> 01:09:51.850 coming from some underlying weight, 01:09:51.850 --> 01:09:55.060 but presumably, this system still, 01:09:55.060 --> 01:09:58.270 if I look at it at long times, will 01:09:58.270 --> 01:10:01.450 have some kind of fluctuations. 01:10:01.450 --> 01:10:04.450 All the fluctuations of this growing surface, 01:10:04.450 --> 01:10:07.340 like the fluctuations of the soap bubble, 01:10:07.340 --> 01:10:10.986 and they have this logarithmic dependence. 01:10:10.986 --> 01:10:13.194 You have a question? 01:10:13.194 --> 01:10:17.400 AUDIENCE: So why doesn't that term-- what if I put in h times 01:10:17.400 --> 01:10:19.968 that term that we want to appear and then I 01:10:19.968 --> 01:10:21.200 vary with respect to h? 01:10:21.200 --> 01:10:24.019 A term like what we want to pop out together with other terms? 01:10:24.019 --> 01:10:25.810 MEHRAN KARDAR: Yeah, but those other terms, 01:10:25.810 --> 01:10:27.660 what do you want to do with them? 01:10:27.660 --> 01:10:32.700 AUDIENCE: Well, maybe they're not acceptable [INAUDIBLE]? 01:10:32.700 --> 01:10:35.360 MEHRAN KARDAR: So you're saying why not have a term that 01:10:35.360 --> 01:10:38.500 is h gradient of h squared? 01:10:38.500 --> 01:10:42.690 Functional derivative of that is gradient of h squared. 01:10:42.690 --> 01:10:47.390 And then you have a term that is h Laplacian-- it's 01:10:47.390 --> 01:10:50.785 a gradient of, sorry, gradient of h. 01:10:57.816 --> 01:11:00.690 And then you expand this. 01:11:00.690 --> 01:11:02.820 Among the terms that you would generate 01:11:02.820 --> 01:11:07.560 would be a term that is h, a Laplacian of h. 01:11:07.560 --> 01:11:09.745 This term violates this condition 01:11:09.745 --> 01:11:13.490 that we had over here. 01:11:13.490 --> 01:11:18.000 And you cannot separate this term from that term. 01:11:18.000 --> 01:11:20.960 So what you describe, you already 01:11:20.960 --> 01:11:22.940 see at the level over here. 01:11:22.940 --> 01:11:27.380 It violates translation of symmetry in [? nature ?]. 01:11:27.380 --> 01:11:29.775 And you can play around with other functions. 01:11:29.775 --> 01:11:31.488 You come to the same conclusion. 01:11:35.300 --> 01:11:39.660 OK, so the question is, well, you added some term here. 01:11:39.660 --> 01:11:43.750 If I look at this surface that has grown at large time, 01:11:43.750 --> 01:11:46.720 does it have the same fluctuations as we had before? 01:11:49.330 --> 01:11:54.820 So a simple way to ascertain that 01:11:54.820 --> 01:11:58.780 is to do the same kind of dimensional analysis 01:11:58.780 --> 01:12:02.570 which, for the Landau-Ginzburg, was a prelude to doing 01:12:02.570 --> 01:12:04.810 renormalization. 01:12:04.810 --> 01:12:08.550 So we did things like epsilon expansion, et cetera. 01:12:08.550 --> 01:12:12.960 But to calculate that there was a critical dimension of 4, 01:12:12.960 --> 01:12:16.300 all we needed to do was to rescale x and m, 01:12:16.300 --> 01:12:18.480 and we would immediately see that mu 01:12:18.480 --> 01:12:21.770 goes to mu, b to the 4 minus D or something-- 01:12:21.770 --> 01:12:24.760 D minus 4, for example. 01:12:24.760 --> 01:12:28.970 So we can do the same thing here. 01:12:28.970 --> 01:12:33.130 We can always move to a frame that 01:12:33.130 --> 01:12:35.250 is moving with the average velocity, 01:12:35.250 --> 01:12:37.910 so that we are focusing on the fluctuations. 01:12:37.910 --> 01:12:40.570 So we can basically ignore this term. 01:12:40.570 --> 01:12:45.620 I'm going to rescale x by a factor of b. 01:12:45.620 --> 01:12:50.980 I'm going to rescale time by a factor of b to something 01:12:50.980 --> 01:12:52.290 to the z. 01:12:52.290 --> 01:12:57.360 And this z is kind of indicative of what 01:12:57.360 --> 01:13:02.430 we've seen before-- that somehow in these dynamical phenomena, 01:13:02.430 --> 01:13:08.620 the scaling of time and space are related to some exponent. 01:13:08.620 --> 01:13:10.690 But there's also an exponent that 01:13:10.690 --> 01:13:14.690 characterizes how the fluctuations in h 01:13:14.690 --> 01:13:18.070 grow if I look at systems that are larger and larger. 01:13:18.070 --> 01:13:20.990 In particular, if I had solved that equation, 01:13:20.990 --> 01:13:24.210 rather than for a soap bubble in two dimensions, 01:13:24.210 --> 01:13:28.650 for a line-- for a string that I was pulling so that I had 01:13:28.650 --> 01:13:31.280 line tension-- the one-dimensional version of it, 01:13:31.280 --> 01:13:35.860 the one-dimensional version of an integral of 1 over q squared 01:13:35.860 --> 01:13:37.940 would be something that would grow 01:13:37.940 --> 01:13:40.100 with the size of the system. 01:13:40.100 --> 01:13:43.100 So there I would have a chi of 1/2, for example, 01:13:43.100 --> 01:13:44.850 in one dimension. 01:13:44.850 --> 01:13:47.250 So this is the general thing. 01:13:47.250 --> 01:13:51.800 And then I would say that the first equation, dhy dt, 01:13:51.800 --> 01:13:54.290 gets a factor of b to the chi minus 01:13:54.290 --> 01:13:58.930 z, because h scaled by a factor of chi, 01:13:58.930 --> 01:14:06.280 t scaled by a factor of z, the term sigma Laplacian of h 01:14:06.280 --> 01:14:10.090 gets a factor of b to the chi minus 2 01:14:10.090 --> 01:14:13.350 from the two derivatives here-- sorry, the z and the 2 look 01:14:13.350 --> 01:14:15.380 kind of the same. 01:14:15.380 --> 01:14:17.412 This is a z. 01:14:17.412 --> 01:14:20.640 This is a 2. 01:14:20.640 --> 01:14:23.830 And then the term that is proportional 01:14:23.830 --> 01:14:27.740 to this non-linearity that I wrote down-- actually, it 01:14:27.740 --> 01:14:32.270 is very easy, maybe worthwhile, to show that sigma to the 4th 01:14:32.270 --> 01:14:36.290 goes with a factor of b to the chi minus 4. 01:14:36.290 --> 01:14:41.100 It is always down by a factor of two scalings in b with respect 01:14:41.100 --> 01:14:44.470 to a Laplacian-- the same reason that when 01:14:44.470 --> 01:14:47.360 we were doing the Landau-Ginzburg. 01:14:47.360 --> 01:14:50.790 We could terminate the series at order of gradient squared, 01:14:50.790 --> 01:14:53.610 because higher-order derivatives were irrelevant. 01:14:53.610 --> 01:14:55.790 They were scaling to 0. 01:14:55.790 --> 01:15:02.890 But this term grows like b to the 2 chi, 01:15:02.890 --> 01:15:08.820 because it's h squared, minus 2, because there's two gradients. 01:15:08.820 --> 01:15:11.950 Now thinking about the scaling of eta 01:15:11.950 --> 01:15:17.670 takes a little bit of thought, because what we have-- we 01:15:17.670 --> 01:15:21.330 said that the average of eta goes to 0. 01:15:21.330 --> 01:15:26.290 The average of eta at two different locations 01:15:26.290 --> 01:15:29.380 and two different times-- it is these particles that 01:15:29.380 --> 01:15:32.460 are raining down-- they're uncorrelated 01:15:32.460 --> 01:15:33.800 at different times. 01:15:33.800 --> 01:15:36.026 They're uncorrelated at different positions. 01:15:38.600 --> 01:15:40.190 There's some kind of variance here, 01:15:40.190 --> 01:15:43.110 but that's not important to us. 01:15:43.110 --> 01:15:48.840 If I rescale t by a factor of b, delta of bt-- sorry, 01:15:48.840 --> 01:15:54.230 if I scale t by a factor of b to the z, delta of b to the zx 01:15:54.230 --> 01:15:57.550 will get a factor of b to the minus z. 01:15:57.550 --> 01:16:02.110 This will get a factor of b to the minus d. 01:16:02.110 --> 01:16:06.240 But the noise, eta, is half of that. 01:16:06.240 --> 01:16:14.210 So what I will have is b to the minus z plus d over 2 times 01:16:14.210 --> 01:16:18.180 eta on the rescalings that I have indicated. 01:16:18.180 --> 01:16:23.780 I get rid of this term. 01:16:23.780 --> 01:16:28.960 So this-- divide by b to the chi minus z. 01:16:28.960 --> 01:16:37.760 So then this becomes bh y dt is sigma b to the z minus 2. 01:16:37.760 --> 01:16:44.550 Maybe I'll write it in red-- b to the z minus 2. 01:16:44.550 --> 01:16:52.470 This becomes sigma to the 4, b to the z minus 4. 01:16:52.470 --> 01:16:54.850 And then lambda over 2. 01:16:54.850 --> 01:16:57.450 This is Laplacian of h. 01:16:57.450 --> 01:17:02.580 This, for derivative of h, this is Laplacian of h. 01:17:02.580 --> 01:17:09.900 This term becomes b to the chi plus z minus 2, gradient of h 01:17:09.900 --> 01:17:12.290 squared. 01:17:12.290 --> 01:17:24.804 And the final term becomes b to the chi minus d minus z over 2. 01:17:36.005 --> 01:17:38.950 AUDIENCE: [INAUDIBLE]? 01:17:38.950 --> 01:17:42.990 MEHRAN KARDAR: b to the minus chi-- you're right. 01:17:42.990 --> 01:17:51.424 And then, actually, this I-- no? 01:17:54.370 --> 01:18:09.944 Minus chi minus d over 2 minus d over 2 [INAUDIBLE], 01:18:09.944 --> 01:18:10.942 That's fine. 01:18:14.940 --> 01:18:21.011 So I can make this equation to be invariant. 01:18:31.813 --> 01:18:34.970 So I want to find out what happens 01:18:34.970 --> 01:18:38.180 to this system if I find some kind of an equation, 01:18:38.180 --> 01:18:41.155 or some kind of behavior that is scale invariant. 01:18:41.155 --> 01:18:45.690 You can see that immediately, my choice for the first term 01:18:45.690 --> 01:18:49.730 has to be z equals to 2. 01:18:49.730 --> 01:18:52.100 So basically, it says that as long 01:18:52.100 --> 01:18:57.060 as you're governed by something that is diffusive, 01:18:57.060 --> 01:19:00.700 so that when you go to Fourier space, you have q squared, 01:19:00.700 --> 01:19:03.430 your relaxation times are going to have 01:19:03.430 --> 01:19:07.140 this diffusive character, where time is distance squared. 01:19:07.140 --> 01:19:10.480 Actually, you can see that immediately from the equation 01:19:10.480 --> 01:19:13.700 that this diffusion time goes like distance squared. 01:19:13.700 --> 01:19:17.380 So this is just a statement of that. 01:19:17.380 --> 01:19:21.940 Now, it is the noise that causes the fluctuations. 01:19:21.940 --> 01:19:29.830 And if I haven't made some simple error, 01:19:29.830 --> 01:19:33.600 you will find that the coefficient of the noise term 01:19:33.600 --> 01:19:36.980 becomes scale invariant, provided that I choose it 01:19:36.980 --> 01:19:41.910 to be z minus d over 2 for chi. 01:19:41.910 --> 01:19:45.610 And since my z was 2, I'm forced to have 01:19:45.610 --> 01:19:48.490 chi to be 2 minus d over 2. 01:19:48.490 --> 01:19:51.770 And let's see if it makes sense to us. 01:19:51.770 --> 01:20:00.170 So if I have a surface such as the case of the soap bubble 01:20:00.170 --> 01:20:04.210 in two dimensions, chi is 0. 01:20:04.210 --> 01:20:06.880 And 0 is actually this limiting case 01:20:06.880 --> 01:20:10.020 that would also be a logarithm. 01:20:10.020 --> 01:20:14.140 If I go to the case of d equals to 1-- like pulling a line 01:20:14.140 --> 01:20:18.090 and having the line fluctuate-- then I have 2 minus 1 01:20:18.090 --> 01:20:20.550 over 2, which is 1/2, which means 01:20:20.550 --> 01:20:22.700 that, because of thermal fluctuations, 01:20:22.700 --> 01:20:26.290 this line will look like it a random walk. 01:20:26.290 --> 01:20:27.710 You go a distance x. 01:20:27.710 --> 01:20:30.420 The fluctuations in height will go 01:20:30.420 --> 01:20:33.290 like the square root of that. 01:20:33.290 --> 01:20:35.000 OK? 01:20:35.000 --> 01:20:37.220 So all of that is fine. 01:20:37.220 --> 01:20:39.660 You would have done exactly the same answer 01:20:39.660 --> 01:20:44.210 if you had just gotten a kind of scaling such as this 01:20:44.210 --> 01:20:46.850 for the case of the Gaussian Model 01:20:46.850 --> 01:20:49.310 without the nonlinearities. 01:20:49.310 --> 01:20:51.850 But for the Gaussian Model with nonlinearities, 01:20:51.850 --> 01:20:55.460 we could also then estimate whether the nonlinearity u 01:20:55.460 --> 01:20:57.210 is relevant. 01:20:57.210 --> 01:21:00.470 So here we see that the coefficient of our nonlinearity 01:21:00.470 --> 01:21:07.690 is lambda, is governed by something 01:21:07.690 --> 01:21:11.260 that is chi plus z minus 2. 01:21:11.260 --> 01:21:17.270 And our chi is 2 minus z over 2. 01:21:17.270 --> 01:21:19.720 z minus 2 is 0. 01:21:19.720 --> 01:21:25.720 So whether or not this nonlinearity is relevant, 01:21:25.720 --> 01:21:28.590 we can see depends on whether you're 01:21:28.590 --> 01:21:31.940 above or below two dimensions. 01:21:31.940 --> 01:21:35.900 So when you are below two dimensions, 01:21:35.900 --> 01:21:39.840 this nonlinearity is relevant. 01:21:39.840 --> 01:21:42.720 And you will certainly have different types 01:21:42.720 --> 01:21:45.810 of scaling phenomena then what you 01:21:45.810 --> 01:21:52.620 predict by the case of the diffusion equation plus noise. 01:21:52.620 --> 01:21:56.300 Of course, the interesting case is 01:21:56.300 --> 01:22:02.120 when you are at the marginal dimension of 2. 01:22:05.840 --> 01:22:12.030 Now, in terms of when you do proper renormalization group 01:22:12.030 --> 01:22:15.820 with this nonlinearity, you will find that, 01:22:15.820 --> 01:22:20.350 unlike the nonlinearity of the Landau-Ginzburg, which 01:22:20.350 --> 01:22:23.800 is marginally irrelevant in four dimensions. 01:22:23.800 --> 01:22:29.250 du by dl was minus u squared, this lambda 01:22:29.250 --> 01:22:31.670 is marginally relevant. 01:22:31.670 --> 01:22:36.320 d lambda by dl is proportional to plus lambda squared. 01:22:36.320 --> 01:22:44.040 It is relevant marginality-- marginally relevant. 01:22:44.040 --> 01:22:46.340 And actually, the epsilon expansion 01:22:46.340 --> 01:22:51.250 gives you no information about what's happening in the system. 01:22:51.250 --> 01:22:55.050 So people have then done numerical simulations. 01:22:55.050 --> 01:22:57.845 And they find that there is a roughness that 01:22:57.845 --> 01:23:02.640 is characterized by an exponent, say something like 0.4. 01:23:02.640 --> 01:23:06.780 So that when you look at some surface that is grown, 01:23:06.780 --> 01:23:12.130 is much, much rougher than the surface of a soap bubble 01:23:12.130 --> 01:23:15.950 or what's happening on the surface of the pond. 01:23:15.950 --> 01:23:21.820 And the key to all of this is that we wrote down equations 01:23:21.820 --> 01:23:24.410 on the basis of this generalization 01:23:24.410 --> 01:23:27.760 on symmetry that we had learned, now applied 01:23:27.760 --> 01:23:31.310 to this dynamical system, did an expansion, 01:23:31.310 --> 01:23:34.640 found one first term. 01:23:34.640 --> 01:23:37.330 And we found it to be relevant. 01:23:37.330 --> 01:23:39.980 And is actually not that often that you 01:23:39.980 --> 01:23:42.460 find something that is relevant, because then it 01:23:42.460 --> 01:23:44.990 is a reason to celebrate. 01:23:44.990 --> 01:23:47.150 Because most of the time, things are irrelevant, 01:23:47.150 --> 01:23:50.260 and you end up with boring diffusion equations. 01:23:50.260 --> 01:23:53.170 So find something that is relevant. 01:23:53.170 --> 01:23:56.010 And that's my last message to you.